On the Location of the Zeros of a Polynomial

In this paper, we obtain a conclusion on where all the zeros of polynomial P(z) = a_nzⁿ +a_n-3z^n-3  + … + a₁z +a_0, where z is a complex variable and a'_ks are the complex coefficients, are located. Precisely, a Ring-shaped region containing all the zeros of polynomial p(z) has been given. In conclusion, along with a few other results that were based on the original Cauchy’s work, sharpens some previously well-known results. Numerous results in this direction have been extended, including various known extensions and generalizations of Cauchy's classical result . This extension is achieved in a fairly uniform manner, enclosing a range of related outcomes. This research not only advances theoretical knowledge but also holds practical implications for fields where polynomial roots play a crucial role. Furthermore, through examples, we demonstrate that our findings offer more insightful information on the roots of polynomials compared to existing results.